3. Harmonically Forced Vibration and Resonance

Introduction and context

The free-vibration models developed earlier describe motion initiated by an initial displacement or velocity and subsequently governed by the system’s own natural characteristics. Engineering systems, however, are rarely isolated from continuing external disturbances. Rotating machines generate periodic centrifugal forces, reciprocating engines produce harmonics of shaft speed, foundations transmit ground motion, and fluid machinery imposes pressure fluctuations. These disturbances can sustain vibration even after the initial transient has disappeared.

The response of a mechanical system to a sinusoidal input is called harmonically forced vibration. Its analysis is central to vibration design because a periodic force may produce a response much larger than the static deflection associated with the same force amplitude. The most important condition occurs when the forcing frequency approaches a natural frequency. This condition, known as resonance, can lead to excessive displacement, fatigue, noise, loss of precision, or structural failure.

This chapter develops the response of the viscously damped single-degree-of-freedom system to harmonic force and harmonic base motion. It distinguishes transient and steady-state behavior, derives frequency-response functions, interprets amplitude and phase, and examines rotating unbalance and vibration isolation. The single-degree-of-freedom model is idealized, but the concepts of dynamic amplification, phase lag, resonance, and transmissibility remain fundamental in multi-degree-of-freedom and continuous-system analysis.

Learning objectives

After studying this chapter, the reader should be able to:

1. Formulate the equation of motion for a damped system subjected to harmonic force or base excitation.

2. Derive and interpret the transient and steady-state responses.

3. Define frequency ratio, damping ratio, dynamic magnification factor, and phase angle.

4. Explain resonance and distinguish its various practical meanings.

5. Construct and use displacement, velocity, and acceleration frequency-response functions.

6. Analyze vibration caused by rotating unbalance.

7. Derive force and motion transmissibility for isolators.

8. Select damping and operating conditions using engineering trade-offs.

9. Identify modeling assumptions, measurement issues, and common analytical errors.

The forced-vibration model

Consider a mass–spring–damper system with displacement x(t), mass m, viscous damping coefficient c, and stiffness k. A harmonic external force is represented by

F(t) = F₀ sin ωt,

where F₀ is the force amplitude and ω is the forcing angular frequency. Newton’s law gives

m x¨ + c x˙ + kx = F₀ sin ωt. (3.1)

The undamped natural frequency and viscous damping ratio are

ωₙ = √(k/m), ζ = c/(2mωₙ),

and the frequency ratio is

r = ω/ωₙ.

Using these quantities, the equation may be written in normalized form as

x¨ + 2ζωₙx˙ + ωₙ²x = (F₀/m) sin ωt.

The static deflection produced by a force of amplitude F₀ is

Xst = F₀/k.

This is a useful reference quantity because the dynamic displacement amplitude can be expressed as a multiple of Xst.

Transient and steady-state components

The complete response is the sum of the complementary, or transient, solution and the particular, or steady-state, solution:

x(t) = xtr(t) + xss(t).

For an underdamped system, the transient component has the form

xtr(t) = e^(-ζωₙt)[C₁ cos(ω_dt) + C₂ sin(ω_dt)],

where

ω_d = ωₙ√(1 − ζ²)

is the damped natural frequency. The constants C₁ and C₂ are determined by the initial conditions and, if the force is switched on suddenly, by the manner in which it is applied.

The exponential factor causes the transient to decay. In a real system, this component may disappear rapidly or may persist for many cycles if damping is small. The steady-state component remains as long as the harmonic force remains active and is independent of initial conditions.

A convenient form for the steady-state response is

xss(t) = X sin(ωt − φ),

where X is the displacement amplitude and φ is the phase lag of displacement behind the applied force. Substitution into the equation of motion gives

X = (F₀/k) / √[(1 − r²)² + (2ζr)²], (3.2)

and

tan φ = (2ζr)/(1 − r²). (3.3)

The phase must be selected in the correct quadrant. For 0 < r < 1, the displacement is generally less than 90° behind the force; at r = 1, the phase is 90°; and for r > 1, it approaches 180° as the frequency becomes very large.

The dimensionless quantity

M(r) = X/Xst = 1/√[(1 − r²)² + (2ζr)²]

is called the dynamic magnification factor or displacement magnification factor. It is the principal amplitude frequency-response function for force excitation.

Interpreting the frequency response

Equation (3.2) provides more than a formula for amplitude. It describes how the balance among inertia, damping, and stiffness changes with frequency.

At low frequency, r << 1, the spring force dominates. Consequently,

X ≈ F₀/k,

and the displacement is nearly in phase with the force. The system behaves approximately as a static spring.

At r = 1, the inertial and elastic terms cancel in the idealized frequency-domain balance, leaving damping to limit the response. The amplitude becomes

X|r=1 = Xst/(2ζ).

Thus, for small damping, the resonant amplitude can be many times the static displacement. The phase lag is 90°.

At high frequency, r >> 1, inertia dominates and

X ≈ F₀/(mω²).

The displacement becomes small and nearly 180° out of phase with the force. Although displacement decreases, velocity and acceleration need not decrease in the same way. This distinction is important when assessing human exposure, bearing loads, or component fatigue.

The displacement, velocity, and acceleration amplitudes are related by

V = ωX,

A = ω²X.

Their frequency-response functions, normalized by suitable static quantities, can therefore display very different trends. A system that appears acceptable when judged by displacement may still have substantial acceleration at high frequency.

Resonance and resonant frequency

In elementary discussions, resonance is often identified with ω = ωₙ. For a damped force-excited system, the frequency at which displacement amplitude is maximum is slightly lower than the undamped natural frequency. Differentiation of M with respect to r gives

r_peak = √(1 − 2ζ²),

provided ζ < 1/√2. The corresponding maximum magnification is

M_peak = 1/[2ζ√(1 − ζ²)].

For small damping, r_peak is close to one and M_peak is approximately 1/(2ζ). The distinction between natural frequency, forcing frequency, and peak-response frequency matters in precision testing and parameter identification.

Resonance can also be defined with respect to velocity or energy transfer. The frequency of maximum velocity amplitude is not necessarily identical to the frequency of maximum displacement amplitude. Therefore, engineers should specify the measured response quantity when referring to a resonant frequency.

The quality factor, commonly used for lightly damped systems, is approximately

Q ≈ 1/(2ζ).

A high Q indicates a sharp, large resonance peak and low damping. A low Q indicates a broader and smaller peak. Bandwidth measurements can be used to estimate damping: for light damping, if ω₁ and ω₂ are the half-power frequencies, then

ζ ≈ (ω₂ − ω₁)/(2ωₙ).

The half-power points correspond to an amplitude equal to the resonant amplitude divided by √2, or a power level one-half of the peak value.

Phasors and frequency-response functions

The steady-state analysis becomes particularly efficient when harmonic quantities are represented by complex phasors. Write the force as the real part of F₀e^{iωt} and the displacement as the real part of Xe^{iωt}. The complex amplitude satisfies

(k − mω² + i cω)X = F₀.

Thus, the receptance, or displacement-to-force frequency-response function, is

HₓF(ω) = X/F₀ = 1/(k − mω² + i cω). (3.4)

Its magnitude is

|HₓF| = 1/√[(k − mω²)² + (cω)²],

and its phase is the negative of the force-to-displacement lag according to the chosen phasor convention.

Other commonly used frequency-response functions are:

• Mobility: HᵥF = V/F₀ = iωHₓF.
• Accelerance or inertance: HₐF = A/F₀ = −ω²HₓF.
• Dynamic stiffness: F/X = k − mω² + i cω.

These functions are not merely alternative notations. Receptance is useful for displacement-controlled performance, mobility is useful in force and velocity measurements, and accelerance is common in experimental modal testing. The phase contains information about whether stiffness, damping, or inertia dominates the response.

A Bode plot displays magnitude and phase against logarithmic frequency. A Nyquist plot displays the complex response directly. Such plots are valuable for comparing experimental data with a model, detecting resonances, and identifying changes caused by looseness, damping loss, or stiffness degradation.

Harmonic excitation from rotating unbalance

Rotating unbalance is one of the most common sources of harmonic vibration. Suppose a rotor contains a small eccentric mass mᵤ at radius e and rotates at angular speed ω. The centrifugal force associated with the unbalance has amplitude

F₀ = mᵤ e ω².

For a machine mounted on a flexible support, the force transmitted to the support and the resulting motion occur at the rotational frequency. Substitution into the force-excitation solution yields

X = (mᵤe/m) r² / √[(1 − r²)² + (2ζr)²]. (3.5)

This expression shows that unbalance response depends on the ratio of unbalanced mass to total vibrating mass, eccentricity, speed ratio, and damping.

At very low speed, the unbalance force is small because it varies with ω². Near resonance, the response may be large despite the small physical unbalance. At very high speed, the displacement approaches a limiting value approximately equal to mᵤe/m, although the force and acceleration may continue to increase. The phase change through resonance is also used in balancing machines to determine the correction angle.

Practical control methods include improving rotor balance, increasing support stiffness when appropriate, shifting the natural frequency away from the operating range, adding suitable damping, and using isolation mounts. A design change should be assessed over the complete speed range, since reducing vibration at one speed can increase it elsewhere.

Base excitation and support motion

Many systems are excited not by a directly applied force but by motion of their support. Let y(t) be the prescribed base displacement and x(t) the absolute displacement of the mass. The spring and damper act according to relative displacement and velocity, so the equation is

m x¨ + c(x˙ − y˙) + k(x − y) = 0. (3.6)

For harmonic base motion,

y(t) = Y sin ωt,

assume x(t) = X sin(ωt − φ) in steady state. In complex form,

X/Y = (k + i cω)/(k − mω² + i cω).

The absolute displacement transmissibility is therefore

Tₓ = X/Y = √[1 + (2ζr)²] / √[(1 − r²)² + (2ζr)²]. (3.7)

At low frequency, the mass follows the base, so Tₓ approaches one. Near resonance, the mass motion may be amplified. At sufficiently high frequency, the mass tends to remain relatively stationary while the base moves beneath it, and Tₓ decreases approximately as 2ζ/r for large r.

A related quantity is the relative displacement z = x − y. The ratio of relative motion to base motion is

Z/Y = r² / √[(1 − r²)² + (2ζr)²]. (3.8)

Relative motion is important in suspension travel, seals, flexible couplings, and clearances. A design that minimizes absolute motion may require substantial relative travel, so both quantities must be checked.

Force transmissibility and vibration isolation

For a force-excited machine mounted on an isolator, the force transmitted to the foundation is the vector sum of spring and damping forces:

F_T(t) = kx + c x˙.

The force transmissibility ratio is

T_F = F_T/F₀ = √[1 + (2ζr)²] / √[(1 − r²)² + (2ζr)²]. (3.9)

For the ideal single-degree-of-freedom model, this has the same magnitude form as absolute displacement transmissibility under base excitation. This similarity should not obscure the different physical inputs and outputs.

Isolation requires T_F < 1. Solving the inequality gives

r > √2.

Thus, operating just above the natural frequency is not sufficient for isolation; the forcing frequency must exceed approximately 1.414 times the natural frequency. Increasing frequency ratio improves isolation, but excessive damping can worsen high-frequency transmissibility because the damper directly transmits force.

Damping presents a design compromise. Larger damping reduces the resonant peak and controls start-up or run-down vibration. However, at high frequency it may increase transmitted force. Small damping improves high-frequency isolation but can produce an unacceptable resonance during passage through critical speed. Practical isolator design therefore considers start-up, shutdown, steady operation, shock loads, allowable travel, thermal effects, and aging—not only the nominal operating frequency.

For equipment that must be protected from floor vibration, the isolation problem is often formulated as base excitation. For equipment that must not disturb its foundation, it is formulated as force transmissibility. In both cases, the selected mount must be characterized by realistic dynamic stiffness and damping, since elastomeric and pneumatic isolators are frequently nonlinear and frequency dependent.

Start-up, shutdown, and the importance of transients

The steady-state solution does not describe the complete behavior during a speed sweep. When a rotating machine accelerates through a critical speed, the forcing frequency changes continuously. The response depends on acceleration rate, damping, initial conditions, and the difference between the excitation frequency and the instantaneous natural frequency. A slowly swept system may achieve a larger near-resonant response than a rapidly swept system because it spends more time near resonance.

Transient amplification can also occur when a force is suddenly applied or removed. The steady-state amplitude may be moderate while the initial overshoot is large. Consequently, design checks should include emergency stops, load switching, impact-like changes in excitation, and control-system transients. A frequency-response curve measured under steady sinusoidal excitation cannot by itself predict all such events.

An illustrative engineering case: vibration in a pump skid

Background

Consider an illustrative centrifugal pump mounted on a steel skid supported by four elastomeric mounts. During operation, an imbalance in the rotating assembly produces a harmonic force at the shaft rotational frequency. The pump speed is variable, and the operating range includes start-up, normal operation, and shutdown.

Situation and problem

Operators observe increased vibration near a particular speed. The initial assumption is that the rotor requires balancing. However, measurements show that vibration is modest at low speed, rises sharply over a narrow range, and falls again at higher speed. The maintenance team must determine whether the dominant problem is rotor unbalance, support resonance, inadequate isolation, or a combination of these factors.

Analysis

A simplified vertical model is first constructed using the combined pump-and-skid mass, the estimated vertical stiffness of the mounts, and measured damping. The predicted natural frequency is compared with the shaft-speed range. A sweep test is then performed using accelerometers on the skid and foundation. The amplitude peak and approximately 180° phase change across the peak indicate a structural resonance. The measured peak frequency is slightly below the undamped natural frequency, consistent with a damped force-excited system.

The response is plotted in terms of frequency ratio. The operating speed at maximum production lies close to r = 1, while the desired isolated operating range should satisfy r > √2. The existing mounts are too stiff for adequate isolation at the normal speed. At the same time, very low damping would make start-up resonance severe.

Approach and solution

Three interventions are evaluated. First, rotor balancing is specified to reduce the excitation amplitude F₀. Second, alternative mounts with lower dynamic stiffness are considered so that the operating frequency lies well above the support natural frequency. Third, moderate damping is retained to limit the resonance during start-up and shutdown. The design is checked for static deflection, permissible mount travel, transmitted force, fatigue of skid welds, and stability under lateral loads.

A revised test is conducted after installation. The resonance shifts to a lower speed, the normal operating point moves into the isolation region, and the foundation vibration decreases. Start-up vibration remains visible but is below the specified limit because the mounts retain sufficient damping.

Outcome and learning points

The case demonstrates why balancing alone may not solve a resonance problem. The unbalance determines excitation magnitude, but the support dynamics determine amplification and transmission. It also illustrates the need to consider the complete operating-speed range, relative mount displacement, phase information, and transient passage through resonance.

Modeling assumptions, limitations, and common mistakes

The single-degree-of-freedom model assumes linear stiffness, viscous damping, a fixed mass, and a single dominant mode. These assumptions are often adequate near one well-separated resonance, but they may fail when modes interact or when mounts, joints, bearings, or materials exhibit nonlinear behavior.

Common analytical mistakes include:

• Confusing Hz with rad/s. The relation is ω = 2πf.
• Using static deflection F₀/k as the actual dynamic amplitude.
• Omitting damping near resonance, where its effect is decisive.
• Treating the forcing frequency as the natural frequency without calculating r.
• Using force-transmissibility equations for a base-excitation problem without identifying the input and output correctly.
• Reporting amplitude without phase, even though phase is essential for diagnosing resonance and balancing.
• Assuming that increased damping always improves isolation.
• Ignoring relative displacement and mount travel.
• Applying steady-state formulas to a rapidly changing-speed transient without qualification.
• Neglecting sensor mass, mounting stiffness, aliasing, or insufficient sampling in experimental work.

Parameter uncertainty is also important. Stiffness may vary with preload and temperature; damping may depend on amplitude and frequency; and the actual excitation may contain several harmonics. In such situations, sensitivity analysis, experimental modal testing, and nonlinear or multi-degree-of-freedom models may be required.

Professional practice, safety, and digital tools

Vibration limits are normally established by equipment manufacturers, industry standards, contractual specifications, or safety regulations. Engineers should not select an acceptance limit solely from a convenient textbook value. The measurement location, sensor type, frequency band, averaging method, and quantity measured—displacement, velocity, or acceleration—must be stated.

Poor vibration design can create fatigue cracks, loosened fasteners, bearing damage, seal failure, excessive noise, and unsafe working conditions. Decisions should therefore account for personnel exposure, environmental effects, maintainability, and the consequences of failure. When vibration data are used to trigger automated shutdowns, alarm thresholds and sensor redundancy should be validated so that both missed faults and nuisance trips are controlled.

Modern condition-monitoring systems use accelerometers, tachometers, digital signal processing, order tracking, and spectral analysis. Order tracking separates components associated with shaft speed from fixed structural resonances. Finite-element models can estimate modal frequencies and mode shapes, while parameter-updating methods compare calculated frequency-response functions with test data. Machine-learning methods may assist anomaly detection, but they do not replace physical interpretation: a change in amplitude may result from speed, load, temperature, sensor placement, or structural modification. Traceable data, calibrated sensors, documented preprocessing, and engineering review remain essential.

Design guidance and emerging directions

A robust forced-vibration study normally follows a sequence: identify the excitation mechanism; define the input and response variables; estimate mass, stiffness, and damping; calculate natural frequencies and frequency ratios; obtain amplitude and phase responses; check resonance crossings and transients; and validate the model experimentally.

Current engineering practice increasingly combines operational measurements with digital twins and reduced-order models. These tools can support real-time estimation of changing stiffness or damping and can predict remaining useful life when integrated with fatigue and condition-monitoring models. Semi-active mounts, variable-stiffness supports, electromagnetic actuators, and active vibration-control systems offer additional means of avoiding or suppressing resonance. Their benefits must be weighed against energy use, control instability, sensor dependence, maintenance requirements, and fail-safe behavior.

The most reliable design is rarely the one that merely minimizes vibration at a single frequency. It is the design that remains acceptable across the full frequency and load range, survives passage through resonance, limits transmitted forces, and retains predictable behavior when components age or operating conditions change.

Chapter Summary

Harmonic excitation produces a response consisting of a decaying transient and a persistent steady-state component. For a viscously damped single-degree-of-freedom system, the displacement amplitude is the static displacement multiplied by the dynamic magnification factor, which depends on frequency ratio and damping ratio. The phase changes from nearly zero at low frequency to nearly 180° at high frequency and is approximately 90° at resonance.

Resonance produces large response when forcing frequency approaches a natural frequency; damping limits the peak and broadens the response. Frequency-response functions such as receptance, mobility, and accelerance express the magnitude and phase relationship between harmonic input and output. Rotating unbalance generates a force proportional to rotational speed squared and can excite a support resonance. Base excitation and force excitation require distinct physical interpretations, although their transmissibility expressions have related forms. Effective isolation generally requires a frequency ratio greater than √2, while damping involves a compromise between resonance control and high-frequency force transmission.

Professional analysis must include operating-speed sweeps, transient passage through critical speeds, relative motion, measurement quality, uncertainty, safety, and realistic mount properties. Digital monitoring and model-based tools strengthen diagnosis when combined with sound vibration theory.

Key Terms

Harmonic excitation: A periodic excitation represented by a sinusoidal function of time.

Transient response: The part of the response that depends on initial conditions and decays with time in a damped system.

Steady-state response: The persistent response remaining after the transient has decayed.

Forcing frequency: The angular frequency ω of an external periodic excitation.

Natural frequency: The characteristic frequency at which a system oscillates in free vibration; for the undamped SDOF system, ωₙ = √(k/m).

Damping ratio: The ratio of actual viscous damping to critical damping, ζ = c/(2mωₙ).

Frequency ratio: The ratio r = ω/ωₙ of forcing frequency to natural frequency.

Dynamic magnification factor: The ratio of dynamic displacement amplitude to static displacement amplitude.

Phase lag: The angular delay of the response relative to the excitation.

Resonance: A condition in which a forcing frequency is near a system frequency and produces an unusually large response.

Receptance: The complex displacement-to-force frequency-response function.

Mobility: The complex velocity-to-force frequency-response function.

Accelerance: The complex acceleration-to-force frequency-response function.

Rotating unbalance: Eccentric rotating mass that generates a periodic centrifugal force.

Base excitation: Vibration imposed through motion of a system’s support.

Transmissibility: The ratio of transmitted response amplitude to excitation amplitude.

Force transmissibility: The ratio of force transmitted to a support to the applied harmonic force.

Isolation region: The frequency range in which transmissibility is less than one, beginning ideally at r > √2.

Quality factor: A measure of resonance sharpness, approximately Q = 1/(2ζ) for light damping.

Half-power bandwidth: The frequency interval between response points at which resonant power is one-half of its maximum value.

Review Questions

Short Answer Questions

1. What are the transient and steady-state components of a harmonically forced response?

2. Define frequency ratio and dynamic magnification factor.

3. Why does the phase lag approach 180° at high forcing frequency?

4. What physical mechanism produces rotating unbalance force?

5. State the ideal isolation condition in terms of frequency ratio.

Long Answer Questions

1. Derive the steady-state displacement amplitude and phase angle for a viscously damped SDOF system subjected to harmonic force.

2. Explain the variation of displacement response in the low-frequency, resonant, and high-frequency regions.

3. Develop the frequency-response functions for receptance, mobility, and accelerance, and explain their engineering uses.

4. Derive the absolute displacement transmissibility for harmonic base excitation and interpret its limiting behavior.

5. Discuss the damping trade-off in vibration isolation and explain why damping that controls resonance may increase high-frequency force transmission.

Analytical / Application Questions

1. A machine has natural frequency 20 Hz and damping ratio 0.05. Determine its frequency ratio and displacement magnification factor when excited at 18 Hz. Interpret whether operation is close to resonance.

2. A rotor has unbalance product mᵤe and operates through a speed range containing the support natural frequency. Explain how amplitude and phase measurements can distinguish rotor unbalance from a support resonance.

3. An isolator must protect a precision instrument from floor motion while also limiting motion during start-up. Propose a design procedure that addresses natural frequency, damping, transmissibility, relative displacement, and transient passage through resonance.

22. Multiple Choice Questions with Answers

Question 1

For a viscously damped SDOF system under harmonic force, the steady-state displacement amplitude is proportional to which quantity?

A. The dynamic magnification factor multiplied by F₀/k

B. The damping ratio multiplied by F₀/k

C. The frequency ratio multiplied by k/F₀

D. The natural frequency divided by the static displacement

Correct Answer: A. The dynamic magnification factor multiplied by F₀/k

Question 2

At forcing frequency equal to the undamped natural frequency, the phase lag of displacement behind the force is approximately:

A. 0°

B. 45°

C. 90°

D. 180°

Correct Answer: C. 90°

Question 3

For small damping, the displacement magnification at r = 1 is approximately:

A. ζ

B. 1/(2ζ)

C. 2ζ

D. 1 − 2ζ²

Correct Answer: B. 1/(2ζ)

Question 4

Which frequency-response function is defined as complex displacement divided by harmonic force?

A. Mobility

B. Accelerance

C. Receptance

D. Transmissibility

Correct Answer: C. Receptance

Question 5

The amplitude of the centrifugal force caused by rotating unbalance is:

A. mᵤeω

B. mᵤe/ω²

C. mᵤeω²

D. mᵤω/e

Correct Answer: C. mᵤeω²

Question 6

For harmonic base excitation, the equation m x¨ + c(x˙ − y˙) + k(x − y) = 0 uses which displacement as the excitation?

A. Absolute mass displacement only

B. Base displacement y(t)

C. Static deflection only

D. Relative displacement as an externally prescribed force

Correct Answer: B. Base displacement y(t)

Question 7

In the ideal SDOF model, force isolation begins when the frequency ratio satisfies:

A. r < 0.5

B. r = 1

C. r > √2

D. r > 1/(2ζ)

Correct Answer: C. r > √2

Question 8

Increasing damping in an isolator generally has which combined effect?

A. It always reduces both resonance and high-frequency transmissibility

B. It reduces the resonant peak but may increase high-frequency transmission

C. It eliminates the natural frequency

D. It makes the system statically rigid

Correct Answer: B. It reduces the resonant peak but may increase high-frequency transmission

Question 9

Why can a slowly accelerated machine experience a larger resonance response than a rapidly accelerated machine?

A. Its mass increases during acceleration

B. It spends more time near the resonant frequency

C. Its damping becomes exactly zero

D. Its forcing force becomes independent of speed

Correct Answer: B. It spends more time near the resonant frequency

Question 10

Which measurement is especially useful for identifying a resonance crossing and balancing information?

A. Phase change together with amplitude

B. Static mass alone

C. Ambient temperature alone

D. The mean displacement with no frequency information

Correct Answer: A. Phase change together with amplitude